Thursday, 12 September 2013

Solving an Equation in Mathematics

where x1,...,xn are the unknowns, and c is a constant. Its solutions are the members of the inverse image

ƒ −1[c] = {(a1,...,an) ∈ T1×···×Tn | ƒ (a1,...,an) = c},

where T1×···×Tn is the domain of the function ƒ. Note that the set of solutions can be empty (there are no solutions), a singleton (there is exactly one solution), finite, or infinite (there are infinitely many solutions).
For example, an equation such as

3x + 2y = 21z

with unknowns x, y and z, can be solved by first modifying the equation in some way while keeping it equivalent, such as subtracting 21z from both sides of the equation to obtain

3x + 2y − 21z = 0

In this particular case there is not just one solution to this equation, but an infinite set of solutions, which can be written

{(x, y, z) | 3x + 2y − 21z = 0}.

One particular solution is x = 0, y = 0, z = 0. Two other solutions are x = 3, y = 6, z = 1, and x = 8, y = 9, z = 2. In fact, this particular set of solutions describes a plane in three-dimensional space, which passes through the three points with these coordinates.

The methods for solving equations generally depend on the type of
equation, both the kind of expressions in the equation and the kind of
values that may be assumed by the unknowns. The variety in types of
equations is large, and so are the corresponding methods. Only a few
specific types are mentioned below; a comprehensive treatment is not
possible.
In general, given a class of equations, there may be no systematic method (algorithm)
that is guaranteed to work. This may be due to a lack of mathematical
knowledge; some problems were only solved after centuries of effort. But
this also reflects that, in general, no such method can exist: some
problems are known to be unsolvable by an algorithm, such as Hilbert's tenth problem, which was proved unsolvable in 1970.
For several classes of equations, algorithms have been found for
solving them, some of which have been implemented and incorporated in computer algebra systems, but often require no more sophisticated technology than pencil and paper. In some other cases, heuristic methods are known that are often successful but that are not guaranteed to lead to success.

Brute force, trial and error, inspired guess

If the solution set of an equation is restricted to a finite set (as is the case for equations in modular arithmetic, for example), or can be limited to a finite number of possibilities (as is the case with some Diophantine equations), the solution set can be found by brute force,
that is, by testing each of the possible values. It may be the case,
though, that the number of possibilities to be considered, although
finite, is so huge that an exhaustive search is not practically feasible; this is, in fact, a requirement for strong encryption methods.
As with all kinds of problem solving, trial and error
may sometimes yield a solution, in particular where the form of the
equation, or its similarity to another equation with a known solution,
may lead to an "inspired guess" at the solution. If a guess, when
tested, fails to be a solution, consideration of the way in which it
fails may lead to a modified guess.

Elementary algebra

Equations involving linear or simple rational functions of a single real-valued unknown, say x, such as

can be solved using the methods of elementary algebra.

Systems of linear equations

Smaller systems of linear equations
can be solved likewise by methods of elementary algebra. For solving
large systems numerically, algorithms are used that are based on linear algebra.

Polynomial equations

Main article: Solving polynomial equations

See also: Systems of polynomial equations

Polynomial equations of degree up to four can be solved exactly using algebraic methods, of which the quadratic formula
is the simplest example. Polynomial equations with a degree of five or
higher require in general numerical methods (see below) or special
functions such as Bring radicals, although some specific cases may be solvable algebraically, for example

4x5 − x3 − 3 = 0

(by using the rational root theorem), and

x6 − 5x3 + 6 = 0,

(by using the substitution x = z1/3, which simplifies this to a quadratic equation in z).

Diophantine equations

In Diophantine equations the solutions are required to be integers.
In some case a brute force approach can be used, as mentioned above. In
some other cases, in particular if the equation is in one unknown, it
is possible to solve the equation for rational-valued unknowns (see Rational root theorem),
and then find solutions to the Diophantine equation by restricting the
solution set to integer-valued solutions. For example, the polynomial
equation

has as rational solutions x = −1/2 and x = 3, and so, viewed as a Diophantine equation, it has the unique solution x = 3.
In general, however, Diophantine equations are among the most difficult equations to solve.

Inverse functions

See also: Inverse problem

In the simple case of a function of one variable, say, h(x), we can solve an equation of the form

h(x) = c, c constant

by considering what is known as the inverse function of h.
Given a function h : A → B, the inverse function, denoted h−1, defined as h−1 : B → A is a function such that

h−1(h(x)) = h(h−1(x)) = x.

Now, if we apply the inverse function to both sides of

h(x) = c, where c is a constant value in B,

we obtain

h−1(h(x)) = h−1(c)

x = h−1(c)

and we have found the solution to the equation. However, depending on
the function, the inverse may be difficult to be defined, or may not be
a function on all of the set B (only on some subset), and have many values at some point.
If just one solution will do, instead of the full solution set, it is actually sufficient if only the functional identity

h(h−1(x)) = x

holds. For example, the projection π1 : R2 → R defined by π1(x, y) = x has no post-inverse, but it has a pre-inverse π1−1 defined by π1−1(x) = (x, 0). Indeed, the equation

π1(x, y) = c

is solved by

(x, y) = π1−1(c) = (c, 0).

Examples of inverse functions include the nth root (inverse of xn); the logarithm (inverse of ax); the inverse trigonometric functions; and Lambert's W function (inverse of xex).

Factorization

If the left-hand side expression of an equation P = 0 can be factorized as P = QR, the solution set of the original solution consists of the union of the solution sets of the two equations Q = 0 and R = 0. For example, the equation

can be rewritten, using the identity tan x cot x = 1 as

which can be factorized into

The solutions are thus the solutions of the equation tan x = 1, and are thus the set

Numerical methods

With more complicated equations in real or complex numbers, simple methods to solve equations can fail. Often, root-finding algorithms like the Newton–Raphson method
can be used to find a numerical solution to an equation, which, for
some applications, can be entirely sufficient to solve some problem.

Taylor series

One well-studied area of mathematics involves examining whether we
can create some simple function to approximate a more complex equation
near a given point. In fact, polynomials in one or several variables can
be used to approximate functions in this way – these are known as Taylor series.

Matrix equations

Equations involving matrices and vectors of real numbers can often be solved by using methods from linear algebra.

Differential equations

There is a vast body of methods for solving various kinds of differential equations, both numerically and analytically. A particular class of problem that can be considered to belong here is integration, and the analytic methods for solving this kind of problems are now called symbolic integration.

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Lifetime Chartered Member" of "The Chartered Institute of Logistics & Transport - India", New Delhi . I am working a Trainer, Tutor and Course Provider for Logistics and Supply Chain Management . Also provide counseling service to students and professionals regarding education, training and job in the field of Logistics and Supply Chain Management .